It is importatntIt is importatnt

1. FEA/CFD for
Biomedical
Engineering
Week 4: Axial
Members and Beams

2. Members Under
Axial Loading
Book: Saeed Moaveni Finite Element Analysis Theory and
application with ANSYS 4th Edition Chapter 4

3. A Linear element
• We will use a steel column which supports the loads from
various floors
• An FEA model can be discretised into 4 elements and 5 nodes
• Floor loading causes vertical displacement of various points
along column

4. A Linear element
• Assuming axial central loading, we may approximate actual
deflection of column using a series of linear functions,
– They describe the deflection over each element / section of column.
• Note that the deflection profile u represents the vertical (not
the lateral) displacement of the column at various points
along the column.

5. Analysing 1 element
• Linear deflection distribution of an element (c1 and c2 – unknown
coefficients)
𝑢𝑢 𝑒𝑒 = 𝑐𝑐1 + 𝑐𝑐2𝑌𝑌
• For each node
𝑢𝑢𝑖𝑖 = 𝑐𝑐1 + 𝑐𝑐2𝑌𝑌𝑖𝑖 𝑢𝑢𝑗𝑗 = 𝑐𝑐1 + 𝑐𝑐2𝑌𝑌
𝑗𝑗
𝑐𝑐1 = 𝑢𝑢𝑖𝑖 − 𝑐𝑐2𝑌𝑌𝑖𝑖 𝑐𝑐2 =
𝑢𝑢𝑗𝑗 − 𝑐𝑐1
𝑌𝑌
𝑗𝑗
𝑐𝑐2 =
𝑢𝑢𝑗𝑗 − 𝑢𝑢𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑐𝑐1 =
𝑢𝑢𝑖𝑖𝑌𝑌
𝑗𝑗 − 𝑢𝑢𝑗𝑗𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖

6. Analysing 1 element
• Linear deflection distribution of an element (c1 and c2 – unknown
coefficients)
𝑢𝑢 𝑒𝑒 = 𝑐𝑐1 + 𝑐𝑐2𝑌𝑌
𝑐𝑐2 =
𝑢𝑢𝑗𝑗 − 𝑢𝑢𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑐𝑐1 =
𝑢𝑢𝑖𝑖𝑌𝑌
𝑗𝑗 − 𝑢𝑢𝑗𝑗𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑢𝑢 𝑒𝑒 =
𝑢𝑢𝑖𝑖𝑌𝑌
𝑗𝑗 − 𝑢𝑢𝑗𝑗𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
+
𝑢𝑢𝑗𝑗 − 𝑢𝑢𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑌𝑌
𝑢𝑢 𝑒𝑒 =
𝑌𝑌
𝑗𝑗 − 𝑌𝑌
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑢𝑢𝑖𝑖 +
𝑌𝑌 − 𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑢𝑢𝑗𝑗

7. Shape Function
• A shape function is a continuous function that is assumed to
represent the approximate physical behaviour (solution) of an
element
• In FEA terms a shape function is the function which
interpolates the solution between the discrete values
obtained at the mesh nodes.
• Appropriate functions have to be used
– Typically low order polynomials

8. Shape Function
• We can define Shape functions Si and Sj for our element
𝑢𝑢 𝑒𝑒 =
𝑌𝑌
𝑗𝑗 − 𝑌𝑌
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑢𝑢𝑖𝑖 +
𝑌𝑌 − 𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
𝑢𝑢𝑗𝑗
𝑆𝑆𝑖𝑖 =
𝑌𝑌
𝑗𝑗 − 𝑌𝑌
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
=
𝑌𝑌
𝑗𝑗 − 𝑌𝑌
𝑙𝑙
𝑆𝑆𝑗𝑗 =
𝑌𝑌 − 𝑌𝑌𝑖𝑖
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖
=
𝑌𝑌 − 𝑌𝑌𝑖𝑖
𝑙𝑙
𝑢𝑢 𝑒𝑒 = 𝑆𝑆𝑖𝑖𝑢𝑢𝑖𝑖 + 𝑆𝑆𝑗𝑗𝑢𝑢𝑗𝑗
𝑢𝑢 𝑒𝑒 = 𝑆𝑆𝑖𝑖 𝑆𝑆𝑗𝑗
𝑢𝑢𝑖𝑖
𝑢𝑢𝑗𝑗
I know you love matrices….

9. Shape Function
• Creating a relationship between the Global and local coordinate
system (we have already seen the advantage of this)
𝑌𝑌 = 𝑌𝑌𝑖𝑖 + 𝑦𝑦 0 ≤ 𝑦𝑦 ≤ 𝑙𝑙
𝑆𝑆𝑖𝑖 =
𝑌𝑌
𝑗𝑗 − 𝑌𝑌
𝑙𝑙
=
𝑌𝑌
𝑗𝑗 − 𝑌𝑌𝑖𝑖 + 𝑦𝑦
𝑙𝑙
= 1 −
𝑦𝑦
𝑙𝑙
𝑆𝑆𝑗𝑗 =
𝑌𝑌𝑖𝑖 + 𝑦𝑦 − 𝑌𝑌𝑖𝑖
𝑙𝑙
=
𝑦𝑦
𝑙𝑙

10. Shape Function
• Si and Sj possess unique properties that can simplify the
derivation of stiffness matrices
• Each have a value of unity (1) at its corresponding node and
zero at the other adjacent node
𝑆𝑆𝑖𝑖 = 1 −
𝑦𝑦
𝑙𝑙
𝑆𝑆𝑗𝑗 =
𝑦𝑦
𝑙𝑙

11. Example 1
Consider a four-story building with steel
columns. One column is subjected to
the loading shown.
E = 200 Gpa
A = 0.026 m2
Determine the deflection of point A and
B when the vertical displacements of
the column at various floor-column
connection points were determined to
be
133kN 133kN
111kN 111kN
111kN 111kN
111kN 111kN
2.44 m
4.57 m
4.57 m
4.57 m
4.57 m
3.04 m
𝑢𝑢1
𝑢𝑢2
𝑢𝑢3
𝑢𝑢4
𝑢𝑢5
= −
0
0.0008338
0.001469
0.001906
0.002144
𝑚𝑚